Read-Shockley Grain Boundaries and the Herring Equation
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1090-Z05-18
Read-Shockley Grain Boundaries and the Herring Equation Shashank Shekhar1, and Alexander H. King1,2 1 School of Materials Engineering, Purdue University, 701 Northwestern Avenue, Neil Armstrong Hall of Engineering, West Lafayette, IN, 47907 2 The Ames Laboratory, 314 TASF, Iowa State University, Ames, IA, 50014 ABSTRACT We compute the strain fields and the interactions between dislocations at the junctions of classical small-angle grain boundaries. It is shown that, in contrast with the results for infinite small-angle boundaries, there are always forces acting on the dislocations in the arrays that define the grain boundaries, and that there is also an excess elastically stored energy associated with the triple junction (TJ). The forces on the dislocations and the excess stored energy of the TJ are shown to vary with the dihedral angles formed by the grain boundaries, and that the “equilibrium” dihedral angle based upon the Herring equation and the energies of the individual grain boundaries does not correspond to any kind of force or energy minimum. This relates to an unwarranted assumption in Herring’s original derivation, that no interactions occur between the grain boundaries that make up a TJ. INTRODUCTION The structures of polycrystals include grain boundaries, triple lines and quadruple points. While a great deal of attention has been paid to the details of grain boundary structure and properties, there has been relatively little work so far on the structure, properties or behavior of the triple junctions that necessarily connect the grain boundary network together. In most cases, the triple junctions are simply assumed to behave in whatever manner is needed to accommodate the grain boundaries that they join. In this paper we address the elastic properties of triple junctions that link grain boundaries, in the framework of classical dislocation theory. Dislocation models for small-angle grain boundaries were introduced by Burgers in 1939 [1] and the energies of such boundaries were calculated by Read and Shockley in 1950 [2]. In the present context it is important to note that the Read-Shockley formula applies strictly to planar boundaries of infinite extent. If such a boundary terminates there is an elastic singularity, and the effects of such singularities at triple junctions form much of the subject matter of this paper and its companion. The effects of surface terminations on interfacial dislocation arrays have been discussed elsewhere [3]. The dihedral angles formed between the grain boundaries at a triple junction are often used in conjunction with the Herring equation [4] to measure the energies of the grain boundaries, and occasionally to relate the grain boundary energy to the misorientation through the Read-Shockley formula [5]. We will show that this may be misleading under certain conditions.
Figure 1: The dislocation configuration considered in our model
MODEL A simple MATLAB® code was written to calculate and minimizes the elastic fields of arrays of dislocations, in the approxi
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